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Error Propagation - Estimating Variance

  1. Oct 29, 2013 #1
    1. The problem statement, all variables and given/known data

    Not exactly a homework question, but rather a section in Statistical Data Analysis:

    Suppose there is a pdf y(x)[/SUB] that is not completely known, but μi and Vij are known:

    2. Relevant equations



    3. The attempt at a solution

    I understand how <y(x)> ≈ y(μ),

    My confusion:

    Why does <y(x2)>

    1. Imply we square everything throughout?

    <[y(μ) + Ʃ[∂y/∂x](xi - μi)]2>

    2. give a xi and xj term? Where did the xj come from?

    3. Why is it for i≠j when xi and xj are uncorrelated, the expression simplifies to

    σ2y ≈ Ʃ[∂y/∂x]2σ2i

    Where did the j go?

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    Last edited: Oct 29, 2013
  2. jcsd
  3. Oct 30, 2013 #2

    Ray Vickson

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    It told you explicitly where the j "went": it said that ##V_{ii} = \sigma_i^2## and that ##V_{ij} = 0 ## for ##i \neq j##.
     
  4. Oct 31, 2013 #3
    Hmm, that makes sense.

    What about the initial derivation? Why did they choose to square the entire RHS when it's a function of (x2) and not f2(x)? And the j's started appearing..
     
  5. Oct 31, 2013 #4

    Ray Vickson

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    Do you honestly mean to say that you cannot tell the difference between ##g(x)^2## and ##g(x^2)##? The paper is working with ##g(x)^2##!
     
  6. Oct 31, 2013 #5
    Ah I see, but where did the j's come from though? And nothing about y(x2) was said that day.
     
  7. Oct 31, 2013 #6

    Ray Vickson

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    Try it for yourself: write out ##(\sum_{i=1}^3 a_i)^2 = (a_1 + a_2 + a_3)^2## in complete detail, by expanding out the square. After doing that, re-write the result using summation notation.

    This will be my last post on this topic.
     
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